I've made this post as detailed as I can so as to give you a fair idea of my level.

Calculus (particularly Integration) is my passion and frankly I spend all my free time learning as much of it as I can. Of late, I've been exposed to several problems on Summations, Integration and Special Functions, as well as the knowledge of theorems such as the Dominated Convergence Theorem, Fubini's Theorem and so on so forth. These problems require greater knowledge of Calculus than what I presently know. This has forced me to search for books on Calculus at higher levels so that I can learn more.

Till date, I have learnt Calculus from thousands of sites on the Internet, and from the following 3 books:

I did as much as I could of Calculus of One Variable a while back and so now have been trying Multi Variable Calculus extensively. I'm quite comfortable (actually done with:)) with Partial Differentiation, Complex Numbers, Trigonometry and to an extent Hyperbolic Trigonometry too. I've done the basics of Multiple Integrals and the basics of some Special Functions (the Beta and Gamma Functions, the Riemann Zeta Function, the Polygamma Functions and the Polylogarithm Functions) too. However, I have never done Linear Algebra or Measure Theory before. I'm also not good at evaluating sums such as Euler Sums.

I would really, really love to to start with Complex methods of Integration such as Contour Integration. However, I've never done it before, so would need a book that starts from scratch.

Under the present circumstances, could you please recommend books which would suitable for me? I'm afraid I tried Apostol, but it was a bit too hard for me. Could somebody suggest books and sources of appropriate standards from where I could study?

I'm most interested in Contour Integration and would really like a book which emphasises it!

  • $\begingroup$ Calculus by Thomas and Finney.... $\endgroup$ – Pratyush Jun 28 '15 at 12:30
  • $\begingroup$ I'm in a similar situation; thank you for the question! May I ask how old you are? $\endgroup$ – Kugelblitz Jun 28 '15 at 12:33
  • $\begingroup$ Welcome Sir! I turned 16 this November. $\endgroup$ – Ishan Jun 28 '15 at 12:35
  • $\begingroup$ @Pratyush Sir, I'm not speaking from personal experience having never read the bok, but a friend of mine said it was quite easy. Is it so, Sir? $\endgroup$ – Ishan Jun 28 '15 at 12:37
  • $\begingroup$ @BetterWorld Sir? I beg to differ; I too turned 16 previous November :D $\endgroup$ – Kugelblitz Jun 28 '15 at 13:05

A delicious classic would be "The Theory of Functions" by Titchmarsh. Despite being old (1939), it is very modern in its rigour, fear not! You may find it freely available on the internet, it is no longer subjected to any copyright.

  • $\begingroup$ Thanks a lot Sir! Sir, could you also please recommend a book where Complex Methods of Integration, especially Contour Integration (and what is required to get started with them) is predominant? I'm really, really eager to start with Contour Integration. You know Sir, every 2 days, I search for on the Net for as many articles as I can get on Contour Integration and then to my disappointment discover that I can't understand a word... $\endgroup$ – Ishan Jun 28 '15 at 17:13
  • $\begingroup$ Sir, I'm just reading through it, and the book seems fantastic! $\endgroup$ – Ishan Jun 28 '15 at 17:20
  • $\begingroup$ @BetterWorld: Instead of searching for books every two days, better arm yourself with lots of patience. One cannot become fluent in English in just a few days; similarly, don't expect do understand complex analysis quickly. As an advice, start reading chapter 3 in Titchmarsh's book and, as soon as you do not understand something, search for it back in the previous chapters. Also, you may safely skip some sub-chapters ("Infinite products", Double series" - most of chapter 1 will not be needed). $\endgroup$ – Alex M. Jun 28 '15 at 17:31

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